/ 



HIBMRY OF CONGRESS. # 

# # 

X jTAo// ,w.s..i., { 

^ . ^ 

! UNITED STATES OF AMERICA, i 



Pr 



^^^ 



' 



MEMORANDA 



ON THE 



STRENGTH OF MATERIALS 



USED IN 



ENGINEERING CONSTRUCTION. 



COMPILED AND EDITED BY 

J. K. WHI LDI N, 

CIVIL ELNGINEER, 

WASHINGTON, D. C. 



NEW YORK : 

D. VAN N^'OSTRAND, 

1860. 
6 



.Hi'' I 



Entered according to Act of Congress, in the year 1860, 

By J. BIEN, 

In the Clerk's Office of the District Court of the United States for the Southern 

District of New York. 



J. A. H. Hasbrouck & Co., Printers, 

Stationers' Hall, New York. 



PHEFACE. 



-•♦^ 



Many valuable accessions to our knowledge of the strength 
of materials have been made within a comparatively recent 
period ; the records, however, have been scattered through 
different books, periodicals, &c. not always accessible or 
convenient for reference. We have therefore endeavored 
in this brief essay to bring together and arrange some of 
the more important elements, both old and new, so as to 
form an " aide memoir^'' for the Engineer and Architect, at 
the drawing office and elsewhere. 

The sources of information have been given as far ag 
possible, in order that the results may be checked, and 
opportunities afforded for lengthened investigations, which 
are beyond the scope of these papers. 

"Wherever it has been necessary to express the strength 
of any material in tons, we have modified the tables and 
formulas to admit the use of the American ton of 2,000 lbs. 
as promotive of simplicity. 



CONTENTS. 

♦♦♦ 

TENSION. 

Tensile strength of timber. 

do do metals. 

do do stones. 

do do wire rope and hempen cable. 

Strength of thin cylinders of cast iron. 
Modulus of elasticity. 
Strength of thick cylinders, as cannon, &c. 
Effects of re-heating, &c. 

COMPRESSION. 

Resistance of timber to crushing. 

do metals do 

do stone do 

Experiments on brickwork, &c, 
Strength of pillars. 
Collapse of tubes. 

DETRUSION. 

Experiments on punching and shearing. 

TRANSVERSE STRENGTH OF MATERIALS. 

Beams of uniform strength. 

Table of co-efficients of timber, stone and iron. 

Relative strength of wrought and cast iron. 

Transverse strength of alloys. 

Table of experiments on cast iron beams of various sections. 

do do wrought iron do do 

Edwin Clark's co-efficients for wrought and cast iron beams, 
LATTICE GIRDERS. 
TRUSSED CAST IRON GIRDERS. 
DEFLECTION OF BEAMS. 
TORSIONAL STRENGTH. 
TORSIONAL ELASTICITY. 



TABLE 

Showing the Tensile strength of Timber, 



Description. 



Weight necesary to tear 






asunder one square inch. 


Modulus of 
elasticity 








Authority. 


In pounds 


In tons of 


in lbs. 




Avoirdupois. 


2,000 lbs, 






17,000 


8.5 


1,600,000 


Barlow. 


6,300 


3.6 




Rankine's Mechanics. 


11,500 


5.7 


1,350.000 


Barlow. 


15,000 


7.5 


1,645,000 


Rankine's Mech. 


20,000 


10. 




Barlow. 


11,400 


5.7 


486,000 


Be van. 


( 10,000 
13,000 


5.0 
6.5 


1,140,000 


Rankine's Mech. 


6,000 


3. 






( 11,000 
\ 11,000 


5.5 


-.. 


Barlow. 


5.780 


2.8 




Do. 


12,000 


6. 




Do. 


8,800 


4.4 






10,500 


5.2 




Bevan. 


18,000 


9. 




Do. 


5,600 


2.8 




Rankine's Mech. 


16,000 


8. 




Bevan. 


20,000 


10. 




Do. 


23.400 


11.7 






7,000 


3.5 




Barlow. 


11,800 


5.9 




Mushenbroek. 


20,580 


10.2 




Barlow. 


16,000 


8. 






\ 8,000 
\ 21,000 


4. 






10.5 


1,255,000 




10,584 


5.2 




Moseley's Mechanics. 


( 10,600 
j 17,400 


5.3 






8.7 






11,501 


5.7 




Barlow. 


9,000 
15,000 


4.5 


1,200.000 


Drt 


75 


1,750,000 




14,000 


7. 




Do. 


14,000 


7. 




Do. 


12,000 


6. 




Do 


14,500 


7.2 




Do. 


9,800 


4.9 




Do. 


10,500 


5.2 




Do. 


11.700 


5.8 




Bevan. 


11,300 


5.6 




Hodgkinson. 


7,200 


3.6 


1,130,000 


Weisbach's Mechanics 


( 5,800 
J 8 800 


2.9 






4.4 






13,000 


65 


1,040,000 


Bevan. 


15000 


7.5 


2,400,000 




8,130 


4.06 




Mushenbroek. 


7,667 


3.8 


820,000 


Moseley's Mech. 


14,000 


7. 




Bevan. 


8,000 


4. 




Do. 


16,626 


8.3 




Moseley's Mech. 



Timber and other orga7iic fibre. 

Ash, (English) 

Bamboo 

Beech, (English) 

Birch 

Box...- 

Cedar 

Chestnut ^'■°°^'- 

to .. 
Cyprus 

DeaiiS'eS!^.:::::::-;: 

Elm 

Fir, or red Pine (New Eng- 
land) 

Do, common American 

Hawthorn 

Hazel 

Hempen cables 

Holly 

Hornbeam (dry) 

Lancewood 

Larch (Scotch) 

Lignum vitae 

Locust 

Do. (Robinia Pseudo A cacia) 

Mahogany 

Maple (Norway) 

Mulberry 

Oak (American white) . ... 
" (English) 

** (African) 

*' (Adriatic) 

** (Canadian) 

*' (Dantzic) 

Pear 

Pine, Pitch 

Plane tree .>..:, 

Plum tree..., 

Poplar 

Quince 

Sycamore..... 

Teak 

Walnut... 

Whalebone 

Willow, dry. ... - 

Yew, Spanish 

Ivory 



TENSILE STRENGTH OF METALS. 



Name of Material. 



"Weight necessary to tear 
asunder one square inch. 



In pounds 
Avoirdupois. 



In tons of 
2,000 lb. 



Modulus of 

elasticity 

in lbs. 



Authority. 



-Antimony, cast , , 

Bismuth, do ',. 

Brass, fine yellow, cast 

Do. Wire 

Copper, cast. ....... 

Do. Sheets ._. 

Do. Bolts 

Do. Wire 

Gun Metal (Coprer andTin) 

Lead, cast (English) 

Do. milled or sheet. ....... 

Steel ...,.- ;... 

Tin, cast 

Do. do ,c.... 

Platinum Wire 

Silver .....; 

Gold, cast , 

Do. Wire 

Zinc 



1,060 

3 250 

17,968 

49,000 

19,000 

30,000 

33,000 

60,000 

36.000 

i;824 

3.300 

100,000 

130,000 

4,600 

5,322 

56,000 

40,000 

20,000 

30,000 

7.000 

8,000 



0.5 
1.6 
8.9 

24.5 
9.5 

15. 

16.5 

30. 

18. 
0.9 
1.6 

50. 

65. 
2.3 
2.6 

28. 

20. 

10. 

15. 
3.5 
4. 



17,000,000 



720,000 
29,000,000 
42,000,000 

4,000,000 



13,000,000 



Muchenbrock. 

Do. 
Rennie. 

Rennie. 

Franklin Institute. 
Rennie. 

Rankine's Mech. 
Rennie. 

Do. 
Tredgold. . 
Rankine's Mech. 

Do. do. 

Rennie 
Mushenbroek. 

U. S. Ordnance 
Manual 1850. 
Mushenbroek. 
Ann. de Chimie XXV 

Rankine's Mech. 



TENSILE STRENGTH OF IRON. 



Cast Iron 

Carron No. 2, cold blast . . . 

Do. hot blast 

Do. No 3, cold blast... 

Do. hot blast , ,. 

Devon No 3, hot blast 

Buffery No. 1, cold blast., . 

Do. hot blast.... 

Coed No. 2, cold blast 

Do. hot blast 

Common Pig Iron 

Good common castings . . . . 

Gun heads, specimens from 

Sterling's toughened Iron. . 



r Salisbury, Conn. 
Bellefonte,Pa. 



^ ! English 



(Q 



Pittsfield, Mass. 
Maramec, Mo.. . 



20,834 


10.4 




Franklin Institute. 




19,200 


96 




Rennie. 




27,700 


13.8 




Cubitt. 




16,683 


8.3 


17,270,500 


\ 




13,505 


6.7 


16,085.000 


) 




14,200 


7.1 


16.246,000 


/ Experiments by 




17,755 


8.8 


17 873,000 


( E.Hodgkinson.for 




29,107 


14.5 


22,473,000 


) the "Rail way Com- 




17,466 


8.7 


15.381,000, I mission" (England) 




13,434 


6.7 


13,730,000 


\ 1849. 




18.855 


9.4 


14,313,000 


/ 




16,676 


8,3 


14,322,000 


/ 




15,000 


7.5 




^ Experiments by 




20,000 


10. 




Maj. Wade for the 




( 24,000 


12. 




[U.S. Ordnance de- 




( 39500 


19.7 




J partment. 




28,000 


14. 




E. Hodgkinson. 


i 


Wrough 


t Iron, 


1 


66.000 


33. •) 




> Franklin Institute. 
Major Wade. 


'.■ 


58,000 


29. 






56,000 
57,000 


28. I 
28.5 f 


29,000,000 




S 43,000 


21.5 






, 


\ 53,000 


26.5 J 









9 

Tensile strength of Iron. 





Weight necessary to tear 
asunder one square inch. 


Mo(1ulus of 
elasticity. 




Name of Material. 


In pounds 
Avoirdupois. 


In tons of 
2,000 lbs. 


Authority. 


WROUGHT IRON, 

Chain Iron 

English rivet Iron - 


43,000 

( 61.000 

j 70,000 

51,000 

35,700 

28,600 

62,640 

56 530 

56,100 

53,900 

142 200 

J 88,600 

\ 134,000 

150,000 

124.000 

1 33,000 

112,000 


21.5 

30 5 

35. 

25.5 

17.8 

14.3 

31.3 

28.2 

28. 

26.9 

71.1 

44.3 

67. 

75. 

62. 

66.5 

56. 


\ 

] 

J 


C.H.Ha3well,Wash. 

Navy Yard. 
"W. Fairbairn. 


Average Boiler plate 

Do. joints double riveted 
Do. do. single do. 

Russian Iron 


Rankin's Mech. 1858. 


English rolled iron 

Lowmoor do. ... - 


United States Board 
of Ordnance. 

Mallet. 


American hammered.. 

Cast Steel (highest) 

Do. (mean) 


Do. tempered .. 

Shear Steel 




Blister Do -. 

Puddled Steel, average of 
three samples 


Mersey Steel and Iron 
Co., England, 1857. 



Brick.. . 
Cement. 
Glass... 

Slate.... 



Natttral and Artificial Stones. 



Mortar Hydraulic, 
Do. common... 



200 




300 




9,4{;0 




9 600 




12.800 




100 to 170 




10 to 50 





8,000,000 VRankine's Mech. 
13 000,000 I 
16,000,000 



Vicat. 
Do. 



Strength of Iron Wire Ro'pe and Hemijcn Cable, 

BY J. A. ROEBLIXG, C. E. 



Circumference' 

of Wire Rope 

in inches. 



Trade 
Number. 



Circumference 

of Hemp Rope 

of equal strength 

in inches. 



Breaking 

weight in tons 

of 2,000 lbs. 



Fine Wire , 



Coarse Wire 



6.62 


1 


15J^ 


74. 


6.20 


2 


\^% 


65. 


5.44 


3 


13 


54. 


4.90 


4 


12 


43.6 


4.50 


5 


105^ 


35. 


391 


6 


^% 


27.2 


3.36 


7 


8 


20.2 


2.98 


8 


7 


16. 


2.56 


9 


6 


11.4 


2.45 


10 


5 


8.C4 


4.45 


11 


10^ 


36. 


4.00 


12 


10 


30. 


3.63 


13 


9Ji 


25. 


3.26 


14 


8k 


20. 


2.98 


15 


7^ 


16. 


2.68 


16 


6^ 


12.3 


2.40 


17 


5^ 


8.8 


2.12 


18 


5 


7.6 


1.9 


19 


4.75 


5.8 


1.63 


20 


4 


4.09 


1.53 


21 


3.3 


2.83 


1.31 


22 


2.80 


2.13 


1.23 


23 


2.46 


1.65 


1.11 


24 


2.2 


1.38 


0.94 


25 


2.04 


1.03 


0.88 


26 


1.75 


0.81 


0.78 


27 


1.50 


0.56 



10 



Practical Rule for finding the Thickness of Cast Iron 
Pipes and other Thin Cylinders, which are exposed to 
Internal Pressure. 

An Extension of the Rule given by J. C. Trautwine, C. E., for water pipes on'y. 

"Most authors who allude to this subject content them- 
selves with merely giving theoretical rules, which are well 
tnown in practice to furnish results entirely too low. Thus 
Barlow's rule gives for a 16 inch water pipe, to support a 
head of 100 feet, a thickness of but yi^ of an inch, or about 
twice that of ordinary letter paper. Other writers again 
give the results of some very incomplete experiments, alto- 
gether too limited in number to serve as general data, while 
others either pass over the subject in silence, or at most 
with an intimation that it admits of no specific instructions. 
# ^ # Mr. Barlow, in giving his rule, assumes the 
safe cohesive strength of cast iron at 18,000 lbs. per square 
inch ; but Mr. Hodgkinson has since conclusively shown, 
by numerous experiments, that it is not safe to employ more 
than 15,000 lbs. per square inch as the ultimate cohesive 
strength of ordinary cast iron. I have assumed 5,000 lbs. 
per square inch as the safe limit for water pipes abstractly 
considered." 

Mr. Trautwine then proceeds to show that a certain 
thickness is necessary to ensure sound casting, and subse- 
quently to bear handling. Now, it is evident that these 
same practical considerations apply to all other cast iron 
cylinders ; those for gas and steam included. 

Hence, with the foregoing data, we furnish the follow- 
ing •— 
Let a=The thickness necessary to ensure good casting, &c-, 

varying according to the values given on next page 
for pipes of different diameters. 



11 

Let F = The pressure in lbs. per square inch on the inner 

surface of pipe. 
Let D=-- The inside diameter of pipe in inches. 
Let T= The entire thickness of pipe in inches. 

ThenT=a + „V^^ - - ' - - No. 1 

Values of a — 

For pipes less than 12 diam. - - - - i^' 

from 12 to 30 " - - - - V 

a^( " *^ 30 *• 48 " - - - - - f " 

cc *^ 48 ** 70 ** - - - - i" 

** 70 '' 100 " - - - - ^" 

Example. — "What must be the thickness of the cylinder 
of a high pressure engine, which is 31 inches bore and ex- 
posed to a pressure of 100 lbs. per square inch ? 

Referring to the table we find a= f ; also, we know 
D=31, and P= 100. 

It is customary in steam cylinders to allow some extra 
thickness for wearing or reboring, so that the above thick- 
ness might be increased to ly'g or IJ. 

THE MODULUS OF ELASTICITY, 

Or the Resistance of Materials to Stretching. 

This is the quotient arising from the division of any 
weight (TF) in lbs, by the relative extension which it pro- 
dujces when suspended from a rod one inch square, or it 
maybe expressed thus: — Extension: length: : weight produc- 
ing extension: modulus of elasticity ; also, representing the 
entire kngth of rod by L ; the extension, or the additional 
length produced by the weight, by ^ / the weight in lbs., 



12 

which produces the extension by W; and the modulus of 

elasticity in lbs. by M, we have as above, e : L :: W : M, 

LW LW 

whence M= , and e= , for rods one inch square. 

e M 

For rods having any other area of cross section, let A re- 
present that area in square inches ; then the above formulas 

LW 
become, il^= No. 2 

Ae 

LW 
and e= No. 3 

AM 

Example. — "What extension will be produced in a bar of 

wrought iron 3 inches diameter and 400 inches long, having 

in suspension a weight of 84,000 lbs. % The average 

modulus of elasticity of wrought iron may be taken at 

28,000,000 \hB,=M; also, A=l square inches, i:=:400 

inches, and W= 84,000 lbs. ; hence, 

LW 400x84,000 33,600,000 12 
e= = = ==.—=0.171 of an 

AM 7x28,000,000 196,000,000 70 

inch. 

Extracts from a Paper on the Strength of Cannon and 
other Thick Cylinders, by Capt. Blakely, K.A. 

(See Civil Engineer and Architect's Journal, for February and March, 1859.) 

** It has long been known that increasing the thickness of 
metal in the cylinder of a Bramah press does not increase 
its strength beyond a certain point. The failure, one after 
another, of the huge presses used in launching the Leviathan, 
under a pressure of less than five tons to the inch, must be 
fresh in the memory of all ; yet the thickness of metal in 
some of these was as much as 7^ inches, the diameter of the 



13 

piston or ram being only 10 inches ; the tensile force of 15 
inches, two thicknesses each 7-^ inches, being thus opposed 
to a pressure on only 10 inches. This small proportion of 
pressure has always been found sufficient to burst a cylinder. 

"The cause of this weakness, which equally affects cannon, 
is even now but very little known, although it was distinctly 
pointed out by Prof. Peter Barlow in a paper read to 
Institution of Civil Engineers many years ago, and pub- 
lished in the first volume of their Transactions. Professor 
Barlow proves that the outer portions of a cylinder have 
comparatively little strain transmitted to them, and therefore 
add little to it« strength. 

*'He bases his argument on the well-known law— that 
metal subjected to a moderate tensile strain, stretches in 
the direction of the strain, and very nearly in proportion to 
it ; that is to say, if one ton will stretch a bar of a certain 
length one inch, two tons will stretch it two inches, or six 
tons six inches. This is generally known as *' Hooke's law," 
nt tensio sic vis, and has been found by experiment to be 
very nearly correct. Prof. Barlow argues that fluid pres- 
sure from within a cylinder must stretch it. That it really 
does so in a gun or rifle is easily proved by placing a leaden 
ring round one ; if it fit tight before the piece is fired, it will 
be-loose afterwards. 

" No known material is capable of exerting a tensile force 
without stretching. The amount of stretching in many ma- 
terials is so well known, that it can be used as a measure of 
the tensile force ; this is the case with iron. 

" Now, a moment's reflection must convince any one that 
the outside of a thick cylinder cannot stretch as much as 
the inside. If it did, the mass of metal or other material 
must be increased during the strain. Let us consider, as 
an example, a cylinder the size and thickness of the strong 
2 



14 

part of a ten-inch gun, ten inches bore, and ten inches of 
metal round it ; and let us imagine this cylinder to be made 
of homogeneous India rubber. In this case the inner diame- 
ter might easily be stretched to double its size, becoming 
twenty inches instead of ten. Now, it is evident that the 
outside circumference and diameter cannot be doubled at 
the same time, or else the latter must become twice thirty, or 
sixty inches, which would give a thickness of twenty inches, 
quadrupling the mass of material — which is impossible. 

"A moment's reflection shows that the thickness must 
diminish as the circumference is increased by pressure 
from within ; for, if the thickness remain 10 inches when 
the internal diameter has become 20, the external diameter 
must be 20 plus twice 10, or 40 inches. This could not be, 
unless wo imagine what seems impossible, viz. : that the 
bulk of the material is considerably enlarged ; as each inch 
in length of the cylinder would now contain 1,200 cylin- 
drical inches (the difference between the squares of 40 and 
20, the external and internal diameters), whereas, originally, 
it only contained 800 inches, the difference between the 
squares of 80 and 10. 

*' Yet even if the thickness could remain the same, 
notwithstanding the increase of circumference, the outside 
layer could only be strained one-third as much as the inside 
one, because three times as long. The same elongation 
which would cause a strain of one ounce or one pound in 
the larger circumference, would cause a strain of three 
ounces or three pounds in the shorter one ; and the elonga- 
tion which would but moderately strain the one would break 
the other. 

*' This reasoning is equally applicable to the minute 
extension of iron ; the increase of one-tenth of an inch in 
the outer circumference of a 10-inch gun being possible 



15 

without fracturing that part, being an elongation of but 1 in 
940 ; whereas the same extension must crack the inside, as 
no iron could stand an elongation of 1 in 314. Even on this 
showing, then, the outside of a thick tube cannot do its share 
of work ; a closer examination, however, must convince us 
that it cannot do even thus much, for the thickness of the 
material must diminish as the circumference is increased. 

*'When the inner diameter of the 10-inch cylinder 
becomes 20 inches, the thickness must diminish from 10 to 
7.32 inches the mass of the material in the cylinder, .and 
therefore its cross section remaining the same. The cross 
section was originally. 800 circular inches — 800 being the 
dijfference between the squares of SO inches, the outer 
diameter, and 10 inches the inner, or 900 minus 100. 
When stretched, the area of the cross section must continue 
to be 800 round inches. 

" Now the thickness of 7.32 inches gives us an external 
diameter of twice 7.32 or 14.64 added to 20, the internal 
diameter ; in all 34.64 inches, the square of which is 1,200, 
subtracting 400 ; the square of 20 leaves 800 round inches 
as before. In this case the outside of the cylinder is 
stretched but 4.64 in 30, about 1 in 7, when the inside is 
stretched to double its original size. 

" If the inner diameter be only stretched to 11 inches, 
the thickness must diminish from 10 to 9.674 inches ; the 
outer diameter becoming 30.348 inches ; the cross section 
remaining 800 round inches as before, the difference be- 
tween the squares of 30.348 and 11. Here the outer layer 
is elongated .348 in 30, or 1 in 86 ; whereas the inner is 
extended 1 in 10, showing a strain or an exertion of force 
8J times greater. 

" In the minute extension of metals, the disproportion is 
still more striking. Thus, in cast iron the 10-inch inner 



16 




diameter may become lOy^Q, which would extend the outer 
diameter only from 30 to SOgi-^, the cross section remaining 
800 inches, and the thickness diminishing from 10 inches to 
9|f|, {Sea Fig, A.) 

Fig. A. 

Here the outside would only be 
stretched 3 1^ 1^ ^0, or 1 in 9,000 ; the 
inside being stretched y^o ^^ 1^» ^^ 1 '^^ 
1000, exerting therefore nine times as 
much force as the outside. 

" It is evident that a slight increase of pressure from 
within would break the inside, while the outside could help 
but little in restraining the disruptive force. 

*^ If we make equi-distant marks on the end of an india 
rubber cylinder (see Fig. B), and stretch it, we can see 
plainly how much more the inside is strained than the out- 
side, or even the intermediate parts. The spaces between 
the marks will become thinner, each space becoming less thin 
than that outside of it, and the inner space much thinner 
than the others {see Fig, C), showing that when the inside is 
strained almost to breaking, the intermediate parts are doing 
much less work, and those far removed almost none. 




Fig. B 



Fig. C. 



17 

" The law deduced by Prof. Peter Barlow, as given below, 
is that in cylinders of metal the force exerted by different 
parts varies inversely as the square of the distances of the 
parts from the axis.'^ 



Frof. Peter Barlow's Formula for the Strength of Thick 

Cylinders, 

" In the first place it is obvious that, whatever extension 
the cylinder or ring may undergo, there will be still in it 
the same quantity of metal ; or, which is the same, the area 
of the circular ring formed by a section through it, will 
remain the same, which area is proportional to the difference 
of the squares of the two diameters. 

" Let D be the interior diameter before the pressure is 
exerted, and D-\-d its diameter when extended by the 
pressure. Let also D' be the external diameter before, 
D'-\-d' the diameter after the pressure is exerted. Then, 
from what is stated above, it follows that we shall have 
B^'—jy^^D'+dJ—iB-^-df, or, 

'lI)'d'-\-d'''=Bd^d\Qr, 

2D'+d' : 2D^-d:\ d \ d' \ 

or, since d' and d are very small in comparison with J^' and 
D, this analogy becomes D' : D \: d \ d'. That is, the ex- 
tension of the exterior surface is to that of the interior as 
the interior diameter to the exterior. 

" But the resistance is as the extension divided by the 
length ; therefore the resistance of the exterior surface is to 

D D' 

that of the interior as — : — or as D^ : D'^ That is, the 

D' D 
resistance offered by each successive lamina is inversely as 



18 

the square of the diameter, or inversely as the square of its 
distance from the centre ; by means of which law the actual 
resistance due to any thickness is readily ascertained. 

*' Let r be the interior radius of any cylinder, jp the 
pressure per square inch on the fluid, t the whole thickness 
of the metal, and x any variable distance from the interior 
surface. Let, also, rp =s represent the strain exerted at inte- 
rior surface ; then by the law last illustrated we shall have 

{r+xy : r'^ :: s : for the strain at the distance x from 

{r+xy 

r^sdx 
the interior surface, and consequently \-Con.—ih.Q 

J {r+xy 

sum of all the strains, or the sum of all the resistance. 

A 1 \ rt 

This becomes, when x=^t, R—rs i - l = s That 

\r r+u r+t 

is, the sum of all the variable resistance due to the whole 

thickness t^ is equal to the resistance that would be due to 

Tt , , 

the thickness , acting uniformly with a resistance ^ 

ovrp.'' 



Application of this Rule for Computing the Proper Thick- 
ness of Metal in a Cylindric Hydraulic Press of given 
Power and Dimensions. 

''Let r be the radius of the proposed cylinder,^:? the 
pressure per square inch on the fluid, and x the required 
thickness. Let also c represent the cohesive strength of a 
square inch of the metal ; then, from what has preceded, it 
appears that the whole strain due to the interior pressure 



19 



will be expressed hj pr, and that the greatest resistance to 

rx 

which the cylinder can be safely opposed is ex ; he-nce, 

r+x 
when the strain and resistance are in equilibrio, we shall 
rx pr 

have rp= Xc, or pr+px=cx, whence x= , the 

r+x c — p 

thickness sought." 



Effects of Re-heating Iro?i, SfC. 

Major Wade's experiments show a very remarkable 
increase in the strength of cast iron when re-heated a 
certain number of times, and maintained in a state of fusion 
some hours previous to pouring; and in a paper read before 
the Society of Arts in England, by Wm. Clay, of the Mersey 
Steel and Iron Works, it was shown that wrought iron also 
increases in strength with each successive piling, heating 
and rolling up to a certain number of times, after which it 
decreases at an equal rate. The following table shows the 
breaking weight in lbs. per square inch of the successive 
samples : — 



1. Puddled bar 43,904 

2. Re-heated 52,864 

3. '' 59,585 

4. *' 59,584 

5. '' 57,354 

6. " 61,824 



7. Reheated 59,585 



8. 

9. 
10. 
11. 
12. 



,57,344 
,57,344 
54,105 
,51,968 
,43,904 



" It will thus be seen that the quality of the iron 
increased up to No. 6 [the slight difference of No. 5 may 
perhaps be attributed to the sample being slightly defec- 
tive), and that from No. 6 the descent was in a similar ratio 
to the previous increase.'' 

Similar experiments on puddled steel bars gave results 
showing a similar increase and decrease. 



20 



EESISTANCE OP TIMBER TO CKUSHING. 



Description. 



Crushing "Weight, 
per square inch. 



In lbs. 



In Tens of 
2,000 lbs 



Weight in 

lbs. of a 

Cubic 

Foot. 



Authority. 



Ash 

Bay wood 

Beech 

Birch, (American) , 

Do. (English) . . 

Cedar 

Red deal ... ..... 

White deal 

Elder 

Elm, Seasoned .... 
Fir, (Spruce) 

Hornbeam ... .... 

Mahogany 

Oak, (Quebec) 

Do. (English) .... 



Do. Dantzic, very dry. . . . 
Pine, Pitch , 



Do. American Yellow 
Do. Red 



Poplar , . . . 

Sycamore 
Teak.... 



Larch r^i;^^^ 

Walnut 



Willow 



8,68.3 

7,518 
7,733 

19,363 
3,297 

11,663 
3,297 
6,402 

'5,674 

5 863 
5,748 

6 586 
6,781 
7,293 
7,451 
9,973 

10,331 
6,499 
6,819 
4,533 
7,289 
8,198 
4,231 
5,982 
6,484 

10,058 
7,731 
6.790 
5,375 
5 445 
5,395 
7,518 
3,107 
5,124 
7,082 

12,101 
3,201 
5 568 
6.063 
7.227 
2.898 
6,128 



4.3 

3.7 

3.8 

9.6 

S1.6 

i5.8 

n.6 

t 32 

i2.8 

/29 

J 2.8 

( 3.2 

)3.3 

/ 3.6 

3.7 

4.9 

5.1 

5 3.2 

)3A 

32.2 

(36 

4.09 
j2.1 
^29 
^3.2 

h. 
3.8 
3.3 
2.6 
2.7 

s 2.6 

} 3.7 

51.5 

\ 2.5 

3.5 

6. 
M.6 
^2.7 
J 3. 
^3.6 
S1.4 
) 3.06 



43. 

53. 

51.3 

53. 

43. 

40.5 

49. 

56. 

47. 

47. 



43. 

36.7 

47. 
50. 
54 5 

58. 

41. 

28.8 

41. 

24. 

43. 
41. 
32. 
35. 

42. 



These are the results 
obtained by Mr.Hodg- 
kinson, from experi- 
ments on short cylin- 
ders with flat ends. 
Each cylinder was 
one inch diameter and 
two inches high. The 
least values in this 
table are for those 
specimens only mod- 
erately dry, and the 
highest, for those 
which were well sea-, 
soned. 

It appears from these 
experiments, that, as 
a general rule, wet 
timber does not pos- 
sess more than half 
the strength of that 
which is well sea- 
soned. 



24. 



10,304 



Heststance 

Brass, yellow 

Cast Iron. 

Buffery, No. 1, cold blast. . . 93 3C6 

Do, hot " 86 397 

Carron, No. 2, cold *' 106,375 

Do. hot " 108^540 

Coed Talon, No. 2, c. b 81,770 

Do. h. b 82J34 

Carron, No. 3, cold blast. . . . 1 15,442 

Do. hot " 133.440 

Devon, No. 3, hot " 145,435 

Stirlings toughened iron 2d i 122!304 

and 3d quality j 1 44,256 

W«oroHTl.o.. j'^T 136,000 



of Metals to Crusliin^. 



5.15 



46.68 
43.19 
53 18 
54.27 
40.-68 
41.36 
57.72 
66.72 
72.71 
61.15 
72,12 



18. 
20. 



Eennie. 



E. Hodgkinson. 
I Experimeuts made 
)-on prisms, base one 

inch square, hight 

1 % inches. 



Rankine's Mechanics. 



I 



21 



RESISTANCE TO NATURAL AND ARTIFICAL 
STONES TO CRUSHING. 



Description. 



Crushing -weight per 
square iuch. 



TVeight in lbs 
of a cubic foot. 



In lb3. 



In tons of 
2.000 lbs. 



Authority. 



Brick (common) 

Do. Fire 

Brickwork 

Chalk 

Granite 

i K ( Barnack ., .. 

e| jChilmark 

■^ tj ( Hamhill (silicious) 

J I rBolsover 

.| o I Huddlestone 

w)| ] Roach Abbey 

g ;S [ Park Nook 

. fAncaster 

I ] Bath Box 

'"o ) Portland 

® LKetton 

B? f Craigleith 

I j Darley Dale 

■^{ Heddon 

c 1 Kenton 

w I Mansfield 

I I f Belleville. N. J. .. 
^S ) Connecticut 

I -§ 1 Dorchester 

I I L Little Fall3 

Caen Stone 

White Marble 

Rubble Masonry 



500 
600 
1,700 
417 
612 

234 

5 500 
11,000 
1 120 
2,912 
1 545 
4,928 
2,3C6 
1,680 
716 
1.6S0 
1,254 
2,128 
1,545 
4,233 
6,160 
L836 
3,382 
1.971 
3,522 
3.319 
3.059 
2,991 
1,088 
5,000 



0.25 
0.40 
0.85 

0.208 
0.3C6 

0.16 

2.75 
5 50 
0.56 
145 
77 
2 46 
1.15 
0.84 
0.35 

84 
0.62 
1.C6 
0.77 
2.11 
3.08 
0.91 
1.69 
0.99 

1 76 
1.65 
1.52 
1.49 
0.54 
2.5 



1.30 
135 

112 

116 
174 
164 
166 
126 
150 
140 
144 
132 
131 
131 
137 
113 
132 
126 
139 
162 
139 
140 
143 
145 
150 
148 
145 
138 
168 



4- 10th of that of cut stone. 



Eennie. 

Eankine's Mech. 
Edwin Clark. 

Eennie. 
Eankine's Mech. 



Experiments made 
on stones for the 
new houses of Par- 
liament, England. 

Note— These weights 
are those ■which pro- 
duced fracture. The ab- 
solute crushing weights 
were occasionally dou- 
ble those here given. 

(Mahan's Civil En- 
gineering, ijth Ed.) 



Experiments bv R. J. 
Hatiicld, xVrch. N. Y. 

(Civil Engineer and 
Architect's Journal, 
May 1858. 

Eankine's Mech. 



Results of Experiments made with Actual Weight on 
Materials used in the Briitania Bridge, January , 1848. 
[E Clark.) 



BRICKWORK. 

No. 1. — 9 incli cube of cGmented brickwork [Nowell & Co.] 
No. 1 (or best quality), weighing 54 lbs., set between 
deal boards ; crushed with 44,654 lbs. = 551 lbs. per 
square inch. 

No. 2, — 9 inch brickwork, No 1, weighing 53 lbs. set in 
cement ; crushed with 49,633 lbs. =612 lbs. per 
square inch. 



22 

No. 3. — 9 inch brickwork, No. 3, weighing 52 lbs., set in 

cement; cruslied with 36,800 lbs. =454 lbs. per 

square inch. 
No. 4. — 9^ inch brickwork, No. 4, weighing 55 J lbs., set in 

cement; crushed with 48,653 lbs =568 lbs. per 

square inch. 
No. 5. — 9 inch brickwork, No. 4, weighing 54 J lbs., set 

between boards ; crushed with 33,836 lbs. =417 lbs. 

per square inch. 
Mean crushing weight per square inch=521 lbs. 
*' The last three cubes of common brick continued to 
support the weight, although cracked in all directions ; they 
fell to pieces when the load was removed. All the brick- 
work began to show irregular cracks a considerable time 
before it gave way. 

" The average weights supported by these bricks was 
37 J tons (of 2,000 lbs.) per square foot, equal to a column 
583 feet high of such brickwork." 

SANDSTONE. 

The average weight required to crush this material is 
150 tons (of 2,000 lbs.) per square foot, equal to a column 
2,351 feet high of this kind of stone. 

LIMESTONE. 

The crushing weight of this material is 527 tons (of 2,000 
lbs.) per square foot, equal to a column 6,433 feet high of 
such stone. 

In the London Builder for 1848, p. 177, there are given 
the results of two experiments on piers of brickwork, 
9 inches square on the plan and 2 feet 3 inches high, 
formed of good sound " Cowley stocks," set in cement mor- 
tar of good quality. One pier was builfc with bricks laid 



23 

flat, the other with bricks on edge ; they were proved two 
days after with the following results : — The pier having the 
bricks laid flatwise compressed one-fourth of an inch, then 
cracked under a weight of 56,000 lbs., and broke to pieces 
with 67,200 lbs. The second one did not compress, cracked 
with a weight of 67^200 lbs., and broke to pieces with 
78,400 lbs. 

Prof. Mahan says that the permanent strain on stone 
should not exceed one-tenth of the crushing weights shown 
by experiments • made on small cubes, measuring about 
2 inches on a side. He also furnishes that part of the fol- 
lowing table which exhibits the ratio of permanent and 
crushing strains on some noted European structures. We 
have appended some additional data obtained from the office 
of the U. S. Lii^hthouse Board. 



Name. 



Permanent 

Strain 
in lbs. per 
square foot. 



Crushing 

weight ia 

lbs. per 

square foot. 



Ratios. 



Pillars of the dome of St. Teter's, (Rome) . 
" " '* St. Pauls'. (London).. 

" " " St. GeneyieVe, (Paris) 

" ** " Church of Toussant, 



(Angers) 

Lower courses of the Bridge of Xeuilly 

^ r U. S. Light House, " 1st Order ;" brick 
tower, 16U feet high 



U. S. Light House, '* 2d Order ;" brick 
tower, 130 feet high 

Minot's ledge Light House, Boston Har- 
bor, Granite tower 

Merchant's Shot tower, (Brick,) Balti- 
more ; 246 feet high, 39 feet 10 
inches base 



33 330 
39,450 
60,000 


536,000 
537,000 
456,000 


1 to 16 
1 to 13.6 
1 to 7.6 


90,000 
3,600 


900,000 
570,000 


1 to 10 
1 to 15.8 


8,300 


71,000 


1 to 8.5 


8,000 


71,000 


1 to 8.8 


7,600 


912,000 


1 to 120 


14,600 


71,000 


1 to4 8 



STRENGTH OF PILLARS. 

Very complete experiments under this head have been 
made by E. Hodgkinson, of Manchester, England. The 
following are some of the more important facts which are 
recorded : — 



24 

*' 1st. In all long pillars of tlie same dimensions, the 
resistance to crushing by flexure is about three times 
greater when the ends are flat than when they are rounded. 

*' 2d, The strength of a pillar with one end rounded and 
the other flat {Fig. 2, see frontispiece) is the arithmetical 
mean between that of a pillar of the same dimensions with 
both ends rounded, and one with both ends flat. Thus, of 
three cylindrical pillars, all of the same length and diameter, 
the first having both its ends rounded [Fig, 1), the second 
with one endrounded and one flat {Fig. 2), and the third with 
both ends flat {Fig. 3), the strengths are as 1, 2, 3, nearly. 

'* 3. A long, uniform cast iron pillar, with its ends firmly 
fixed, whether by means of discs or otherwise, has the same 
power to resist breaking as a pillar of the same diameter 
and half the length, with the ends rounded or turned, so 
that the force would pass through the axis. 

'* 4th. The strength of a cast iron pillar, made like an 
English * connecting rod ' (cruciform section +), was found 
to be very small, perhaps not half as strong as if cast in the 
form of a uniform hollow cylinder. 

*' 5th. A pillar irregularly fixed, so that the pressure 
would be in the direction of the diagonal, is reduced to one- 
third of its strength." 

The following are Mr. Hodgkinson's formula for the 
ultimate strength of pillars, in which 
D — The external diameter, or side of the square, of the 

column in inches. 
d — The internal diameter of hollow column in inches. 
L — The length of column mfeet, 
W — The breaking weight in tons of 2,000 lbs. 



25 



Nature of Coluren. 



Both ends being round- 
ed, the length of the 
column exceeding fif- 
teen times its diameter. 



Both ends being flat, 

the length of the column 

exceeding thirty times 

its diameter. 



Solid cylindrical column 
of Cast Iron 

Hollow cylindrical col- 
umn of Cast Iron 

Solid cylindrical column 
of Wrought Iron 

Solid square Pillar of 
Dantzic oak (dry) 

Solid square Pillar of 
red deal (dry) ... 



TV3.76 



W=14.5 

W=47.9 



'jTt 

J)S.78 



W=49.4 



D 



,3.55 



W=49.6 



P3.55^7 3.55 



W=149.7 



7)3.55 



W=12.2j;— 



W=8.7 



D^ 



*' The above formulas are correct only when the length is 
from 30 to 90 times the diameter, when the colums yield 
wholly by bending." 

" When the columns are shorter than given in the tables, 

or when the material yields by crushing and bending, the 

following formula was found to represent the strength with 

sufficient accuracy. Let ^>=the breaking weight as com- 

13uted by the formula for long columns; let c=the crushing 

be 
weight of the material. Then will W= ." 

^* b+ic 

"The strength of similar pillars- is nearly as the area of 

the cross section." 

Relative Strength of Long Columns of Cast and Wrought 
Iron, Steel and Timber, of the same Dimensions, repre- 
senting the strength of Cast Iron by 1,000. 

C^astiron, 1^000, 

Wrought iron, .... 1,745. 

Cast steel, - - . » . 2,518. 

Dantzic oak, - - . - . 108.8 

Red deal, - . . . - 78 5 

3 



26 

" Wet timber was found not to be one-half as strong as 
whon dry." 

When wrought iron tubes or cells, formed of plates con- 
nected together with angle iron, are used as struts or pillars, 
Messrs. Fairbairn and Hodgkinson state that for a single 
cell, when the thickness of the plates is not less than one- 
thirtieth (3^) of the diameter, the ultimate resistance to a 
compressive strain is 27,000 lbs. per square inch of section ; 
but when a number of cells exist side by side in one girder, 
it may be taken at from 33,000 to 36,000 lbs. per square 
inch. 



The formulas given below are from Professor Eankine's 
Applied Mechanics, 1858 ; they apply to Pillars of any 
length or any form of cross section. 

Let W=The weight or pressure on the pillar in lbs. 

" A=The cross sectional area in square inches. 

** L=The total length of pillar in inches. 

" A=The diameter if round, or the least dimension across 

the section, if otherwise shaped. 

" y=A co-efficient of the material in respect to compression 

*' a =do. do. do of flexure. 

f A. 
Then for pillars having flat ends, W=i No. 4. 

For pillars with rounded ends, W= -^ No 5 

i+Hi)' 

The value of a for wrouo-ht iron= — 

o 3,000 

And for cast iron a = — 



27 



The values of (^f) for different kinds of loads are asfolknvs : 





BreakiDg Load. 


Proof Load. 


Working Load. 


Wrought Iron 


36,000 
80,000 


18,000 
26,700 


6,000 to 9,000 
13,300 *' 20,000 


Cast Iron . 





Tables ezhibiti7ig the comparative strengths of lorought and 
Cast Iron Pillars of various lengths. 

PiLLASS WITH FLAT ENDS. 

Breaking load in lbs. per square incli of the cross section. 



L_ 

h 


10 


20 


26.4 


30 


40 


50 


60 


70 


80 


"Wrought Iron 

Cast Iron 


34840 
64000 


31765 
40000 


29230 
29230 


27700 
24620 


23480 
16000 


19670 
11000 


16300 
i 8000 


13800 
6060 


11600 
4700 







The above table was coniiDuted from formula No. 4. 

PiLLAPvS WITH ROUNDED ENDS. 

Breaking load in lbs. per square inch of cross section. 

(Computed from formula No. 5.) 



L _ 

h 


10 


13.15 


20 


30 


40 


50 


60 


70 


80 


Wrought Iron 

Cast Iron 


31850 
40000 


29200 
29200 


23520 
16000 


16260 

8000 


11500 

47C6 


8370 
3000 


6200 
2100 


4800 
1600 


3780 
1200 



The strength of connecting rods of steam engines, and 
other similar appliances, where the ends of the rods are 
moyable, may be estimated from the latter table. 

Example. — ^What is "the breaking weight of a wrought iron 
connecting rod 16 feet 8 inches long, having a diameter in 
the centre of 5 inches ? Here L = 20C, and h= 5" 
then -j^= -^ = 40, and looking in the table for the corres- 
ponding number, we find the breaking weight per square 
inch to be 11500 lbs. Now the area of a rod 5 inches in 
diameter, is about 19.6 square inches, hence the entire 
breaking weight is 11500 w 19.6=225,400 lbs. If we assume 
the safe weight to be \ of the breaking weight, we have 
— - — =56350 lbs. at the load which may be borne with 
safety. 



28 



COLLAPSE OF TUBES. 



The practical rule at present much used is, where tubes 
are exposed to external pressure, to double that thickness 
which would be necessary for internal pressure. 



Experiments made by Wm. Fairhairn, for the British 
Association, 1857, on the Resistance of Tubes {ii'on) to 
Collapse, 

So far as these experiments have been conducted they 
indicate, 1st, that the strength of tubes of equal diameter 
and thickness varies inversely as their length ; that is to 
say, if a tube one foot long sustains a given pressure, the 
same tube, if two feet long, would bear but one-half that 
pressure. 

2d. Tubes of the same length and thickness, but of 
different diameters, the strength is inversely as the 
diameter. ^ 

This latter rule agrees with the ordinary and accepted 
theory on this subject. 

The law for the thickness seems not to have been 
determined as yet. 

In all the tubes experimented upon, the ends were 
secured in a manner somewhat analogous to the modes 
adopted in steam boilers, and it seems reasonable to con- 
clude that this fact has much to do with the remarkable 
results obtained. 



29 



Table of Experiments on the Collapse of Tubes, 

(The tubes collapsed with a loud report and a hissing noise.) 







Diameter 








Pressure 






No. of 
Experi- 
ment. 


of the 
tube in 
inches. 


Length 
Tube. 


Thickness of 

Tubes 
in inches. 


of colapse 

in lbs. per 

sq. in. 




The discrepancies in 


1 


6 


2' 


6" 


.043 


48 


\ 


experiments from Xo. 1 


2 


6 


2' 


5" 


.043 


47 


/ % 


to No. 6, are due mainly 


3 


6 


4' 


11" 


.043 


32 


\ S" 


to the modes of securing 


4 


6 


2' 


6" 


.043 


52 


( & 


the tubes while being 


5 


6 


2' 


6" 


.043 


65 


1 w 


tried. 


6 


6 


2' 


6" 


.043 


85 


J ' 




7 


4 


1' 


7" 


.043 


170 


\ 




8 


4 


1" 


V 


.043 


137 


/ *^ 




9 


4 


3' 


4" 


.043 


65 


\ rt' 




10 


4 


3' 


2" 


.043 


65 


i 5* 


The tube in experiment 


11 


4 


6' 


0" 


.043 


43 


) ^ 


No. 12 may be considered 


12 


4 


5' 


0" 


.043 


140 




as three distinct tubes, 


13 


8 


2' 


6" 


.043 


39 


?^« 


as it had two rigid rings 


14 


8 


3' 


3" 


.043 


32 


cl = 


soldered to the outside, 


15 


8 


3' 


4" 


.043 


31 


) " 


the result of this simple 


16 


10 


4' 


2" 


.043 


19 


i&s 


contrivance was to in- 


17 


10 


2' 


6" 


.043 


33 


S s * 


crease the strength three- 


18 


12.2 


4' 


103^" 


.043 


11 




fold. 


19 


12 


5' 


0" 


.043 


12.5 






20 


12 


2' 


6" 


.043 


22 





From experiments 21 and 22, on elliptical tubes, it will 
be seen how great is the reduction of strength when we 
deviate from a circular form, by comparing experiment 22 
with experiment 23, it appears that the cylinder in this ease 
han more than three times the strength of the elliptical tube, 
under otherwise similar conditions. 



Elliptical tubes. 

Tubes with lap 
and butt joint 



Number 

of 
Experi- 
ment. 



21 
22 
23 
24 
25 
26 



Diameters of 
tube in 
inches. 



Length 
of tube. 



Thickness! Pressure of 



of Tubes 
in inches. 



collapse in lbs. 
per sq. in. 



14"xl0i^" 

18^ 
9 
9 
9 



3' 1" 
3' 1" 



,043 
.25 
.25 
.25 
.14 
.14 



6.5 
127.5 
420. 

262, 
378. 



30 



Resisience of Tubes to internal Pressure, 

The tube tried in experiment 24 was not collapsed, owing 
to the deficiency of strength in the apparatus used, (a cast 
iron cylinder.) Mr. Fairbairn states that it was not capable 
of sustaining with safety over 500 lbs. per square inch. 





Number 

of 
Experi- 
ments. 


Diameters of 
tube in 
incbes. 


Length 
of tube. 


Thickness 
of tubes 
in inches. 


Bursting 

pressure 

in lbs. 

per sq. in. 




27 
28 
29 
30 
31 


6 
6 
6 
6 
12.13 


4' 0" 
2' 6" 
2' 0" 
V 0" 
5' 0" 


.043 
.043 
.043 
.043 
.043 


375 
230 
235 
475 
110 



Punching and Shearing Wrought Iron. 

The following experiments are recorded in the London 
Artiza7i, for 1858 : They were made with an hydraulic 
press, which was constructed for cutting up scrap iron by 
direct pressure. 

Table of experiments on Punching, 



No. of 


Diameter of 

Punch 

in inches. 


Sectional area cut. 


Pressure on Punch 
in tons of 2,000 lbs. 


Bemarkai 


ment. 


Thickness and 
Circumference 


Area in 
sq. in's. 


Total. 


per sq. inch 
of area cut. 


1 
2 
3 
4 
5 
6 


1 
1 
2 
2 
2 
2 


0.51x3.14 
0.98x3.14 
0.52x6.28 
0.57 X 6.28 
1.06x6.28 
1.52H6 28 


1.60 
3.08 
3.27 
3.58 
6.66 
9.55 


40. 
77.6 
66.9 
78.9 
148.7 
209. 


25. 

25.3 

20.4 

22. 

22.2 

21.8 


25.2 mean. 
-21.7 mean. 



31 



Table of experiments on Shearin, 



No. of 


Direction 

of 
Shearing. 


Sectional area cut. 


Pressure on cutters in tons 
of 2000 lbs. 






experi- 
ment. 


Thickness and 


Area in 
square 
inches. 


Total. 


Per square inch 


Remarks. 






breadth in inhs. 




of area cut. 




7 


Flat 


0.50 H 3.00 


1.50 


37.4 


24.9 


j 25 4 mean. |^ 


a 


8 


Edge 


0.50 M 3.00 


1.50 


38.7 


25.8 


2 


9 


Flat 


1.00 X 3.00 


3.00 


77.6 


25.8 


1 g 


3.9 


10 
11 


Edge 
Flat 


].00 « 3.00 
1.00 M 3.02 


3.U0 
3.02 


76.3 
66.9 


25.4 
22.1 


V 24.0 mean. ? 


12 


Edge 


1.00 M 3.02 


3.02 


69.5 


23.0 


J s 


■3" 


13 


Edge 


1.80 M 5. 


10.2 


235.8 




Flanged tyre. 1 

.a 


1 



14 
15 

16 
17 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 



Flat 

Edge 

Flat 

Edge 

Flat 

Edge 

Flat 

Edge 

Flat 

Edge 

Flat 

Edge 

Flat 

Edge 



0.56 X 3.00 
0.56 X 3.00 
0,90 X 3.37 
0.87 X 3.32 
1.06 X 3.02 
1.06 ^ 3.02 

1.52 m 3.03 

1.53 X 3 03 
1.39 X 4.50 
1.38 M 4.50 
1.73 ^ 5.30 
1.73 X 5.30 
1.56 M 6:00 
1.56 X 6.00 



1.68 


23.7 


1.68 


37.1 


3.03 


30.7 


2 89 


64.2 


3.20 


56.6 


3.20 


75.6 


4.61 


93.7 


4.64 


104.4 


6.25 


100.4 


6.21 


124,5 


9.17 


171.5 


9.17 


231.8 


9.36 


156.7 


9.36 


192.9 



14.1 
22.0 
10.0 
22.1 
17.5 
23.6 
20.3 
22 '^ 
16. (f 
?0.0 
18.7 
25.3 
16.8 
20.6 



^ _■ 



bo 



28 
29 
30 
31 
32 



Square 

Square 

Flat 

Edge 

Edge 



3.10^3.10 
3.10 x3.10 
1.80 X5.00 
1.80 H5.00 
1.70 X5.25 



9.61 


184.9 


9.61 


174.1 


10.20 


111.1 


10.20 


207.7 


10.57 


201.0 



19.2 


Hammered Iron 


18.1 


Rolled Iron. 


10.8 


Flanged tyre. 


20.3 


Do. do. 


19. 


Do. do. 



The table of experiments given below were made Iby 
means of weights applied to a pair of levers, giving a total 
leverage of 60 to 1. The weights were added gradually by 
a few lbs. at a time, until the hole was punched. 

Experiments on FuncJiing Plate Iron, [Rolled] 



Diameter of 
hole in 
inches. 


Thickness 

of plate 

in inches. 


Sectional area 

cut through 

in square inches. 


Total pressure 

on punch in tons 

of 2,000 lbs. 


Pressure per 

square inch of area 

cut in tons of 

2,000 lbs. 


0.250 
0.500 
0.750 
0.875 
1.000 


0.437 
0.625 
0.625 

0.875 
1.000 


0.344 
0.982 
1.472 
2.405 
3.142 


9.2 
29.7 
38.8 
62.1 
86.3 


27.3 
30.4 
26.4 
25.8 
27.5 



" The resistance of wrought iron to shearing may be esti- 
mated at from 70 to 80 per cent, of the resistence to direct 
tensile strain." 



32 

TRANSVEESE STEENGH OF MATEEIALS. 

Various theories have, from time to time, been proposed 
for the transverse strength of materials, but as yet scarcely 
any one has been entirely satisfactory or in strict accordance 
with the results shown by experiments. The ordinary and 
least objectionable one is that which assigns a neutral axis 
at the centre of gravity of the cross section of the beam, 
and supposes the material compressed on the concave side 
of beam and extended on the convex side when deflected 
by weight ; the resistance to compression or extension of any 
element being directly as its distance from the neutral axis. 

This theory has been variously modified by Gralileo, 
Lebnitz, Hooke, Dr. Young, and others^ More recently 
Prof. W. H. Barlow has shown that in a beam subjected to 
transverse strains, in addition to the resistances of extension 
or compression, there exists another element which he terms 
resistance of flexure. 

Numerous experiments have been made by eminent 
engineers giving results which are generally, in ordinary 
construction, sufficient to guide our practice. Some funda- 
mental theorems have been abundantly confirmed — such, 
for instance, as that in beams of equal length and breadth, 
the strength is as the square of the depth ; and in 
beams of equal length and depth, directly as the breadth ; 
so that the strength of a beam may be represented by 

breadth X square of depth , . ,. , , ^ . 
r""~7i; — multiplied by a co-euicient, varying 

according to the nature of the material. 

For purposes of comparison, the latest authorities intro- 
duce into the formula of strength the total area of cross sec- 

-^, . , ^ .. breadth X square of depth 

tion. Thus, instead of writmo; —J: — ±- — x 

° length 

co-efflcient, they write ^ ; — - — X co-efficient, and then 

•^ length 



33 

the comparative strength of beams having various proportions 
of cross section, becomes manifest by a simple inspection of 
the co-efficients, which are determined by experiment. 

Messrs. Hodgkinson and Fairbairn, in giving the results 
of their experiments on iron beams, have only introduced 
the area of the bottom flange as an element of calculation. 
Mr. Edwin Clark, in his work on the Britannia and Conway 
bridges, adopts the total area of cross section as a basis of 
calculation ; also, S. Hughes, C.E., who has written a series 
of valuable papers on the strength of beams for the London 
Artizan Journal, 1857 and '58. 

In applying the different formulas for the strength of 
beams, which are given in this article — - 
Let i=The length of the beam in feet, clear of supports. 
Let 7l=^The total area of the cross section in square inches. 
Let J=The depth of beam in inches. 
Let 5=The breadth of beam in inches* 
Let ^•=The co-efficient of breaking weight. 
Let S=The co-efficient of safe weight. 

TFwill be the breaking weight in lbs. or tons, according 
as k is taken in lbs. or tons. If S be substituted for k, the 
result, "H^will be the safe weight. 

Figures 4, 5, 6 and 7, on the lithograph sheet, illustrate 
how the strength of a beam is affected by placing the 
weights and supports differently. The number of balls 
representing the amount of weight ; those shown as shaded 
represent the amount of weight which can be borne when it 
is concentrated in one place, and those shown with dotted 
lines, the amount which can be borne when the weight is 
uniformly distributed. 

The CO- efficient k, as given in the tables, is the breaking 
iveight of a beam \foot long, 1 inch deep, and 1 square inch 
area of cross section {fixed at 07ie end and loaded at the 
other. ) 



34 

The breaking weights of beams variously circumstanced 
are as follows: — 

Fixed at one end, and ) ^^^ hAcl 

, , , , \ \W=^ or PF= - No. 6 

loaded at the other, \ j^ j^ 

Fixed at one end, and | ^J^^f\ or 1^=2/— ^^'°- 
uniformly loaded, \ XL/ \ L / 

Supported at both) / kbd\ /kAcl\^ 

ends, and loaded at [ W=d lor 1^=41 1^* 

the centre, ) \ L ^ \ L / 



Supported at both ) / kh(P\ / kAd\ ^ 

ends, and uniformly \ TT^Sf |orTr==8| \^^' 

loaded, ) \ L / \ L / ^ 

Fixed at both ends, ) / khd\ / kAd\ -sr 

and loaded in the [ W=Qi lorTF=6| T^J' 

centre, as in Fig. 7, ) \ L / \ L / 

Fixed at both ends,) /kbd\ /kAd\yrr 

and uniformly load- } W=12i |orTr=12| — - j^^^' 

ed, as in Fig. 7, ) \ L / V i / 



Beams of uniform strength. 

In order that the strength may be uniform throughout a 
beam, it is obviously necessary that the cross section at dif. 
ferent places should be proportioned to the amount of strain 
upon it. This strain is always directly as the weight multi- 
plied by the length of the arm or lever with which it acts» 
and is what is usually termed by writers on mechanics, the 
** moment of rupture." Hence, when we know the moment 
of rupture for any point on a beam, we can at once determine 
the proper dimensions at that place ; presuming the co-effi- 
cient of the material known. 

Case 1st. — Beam fixed at one end and loaded at the other. 
Here, the moment of rupture isr== TFm L, and the strength 



35 

of the beam [which should of course bo equal to the strain 

u^ou it,]=kbd^=WL - - • Equation, No. 12 

To ascertain how the breadth must vary when the other parts 

W 
remain constant, we find the value of b= m L — No. 13. 

kd' 

W 
as b is to be found for various lengths, while — — remains 

constant, it will be seen that this is an equation to a straight 

line, the breadth being directly as the length. See Fig, 9. 

When the breadth is constant, we find the value of 

^=/F^=II-/^ ■ • • "»■" 

That is, the depth varies as the square root of the length, 
and No. 14 is an equation to the common parabola, with the 

vertex at the free end of beam, [See Fig 8,] the parameter 

W , 

being — . If we wish to determine the ordinate s d, for dif- 

fC 

ferent lengths or distances from the weight, we extract the 

W 

square root of — as already indicated, and as the quantity 

thus obtained remains constant, we multiply it into the dif- 
ferent square roots of L, and the results are the correspond- 
ing ordinates. 

Example. — -A cast iron beam fixed at one end and loaded 
at the other is to be 3 ft. long, and have a uniform breadth 
of 1 inch. The breaking weight to be 1,100 lbs. What 
must the values of d, at the distances J ft., 1 ft., IJ f t , 2 ft.> 
2 J ft. and 3 ft. from the weight 1 

Here Pr=l,100, b=l and k for average, cast iron may be 
taken equal to 550. 



Hence /2^_fil^/2=1.41. 
\kb V550Miy 



36 



*j 



Now when L=3' ft. d=1.4l /S = 2.43" 
" •<• 2|-ft. d=1.4l/2r5=2.22" 
" " 2 ft. (1=141/2=2" 
" " li ft. d=1.4l/r5=1.72'' 
" " 1 ft. d=1.4l/i=141" 
" " i ft. d=1.4l/5=l. 

" • 'r ir ff nf a 

y2 ^ 'ijZ^ //2 X -^2 X i/^x i^ > 



IT 










Wheii the cross sections of the beam are all similar, as 
when they are circles, squares, elipsis, etc., or when the 
breadth bears a constant ratio to the depth, then, expressing 
this ratio by n, we may put b^nd and substituting this value 
in equation No. 12 we find WL=-k{7id)d'^=-knd\ 



Whence d-^ 



fWL 

ij kn 






No. 15. 



That is, d varies as the cube root of the length and as the 

breadth bears a constant ratio to the depth, the curves on 

all sides are similar and are cubic parabolas See Fig. 10. 

Case 2d. — Beam fixed at one end and tmiformly loaded. 




M K/?. E. 



o 



7 



By inspection of Fig. E, it will be seen that any section, a^ 
-/ will only be strained by the weights which lie beyoaS 
it. If we represent by q the weight per unit of length, tfien 
the total weight for any distance x from the free end of beam 
will be qx, and as this weight acts through its centre? oC 

X 

gravity at a distance—;' ^The moment of rupture will be 

qx M ^=^i=JibcP Na. m 

■^2 2 

When the breadth is constant, 

'^l—^ = iwb''^ - - - - No. IT. 

That is, d varies directly as the length or distance froia 
the free end of beam, and consequently the line obtained by 
different values oi d, is straight. See Fig. 11* 

When the depth is constant. 

6= 777-,, ^x"" No. IS- 

Here b varies as the square of the length or distance from^ 
free end of beam, and the curve thereby produced is the 
common parabola with the vertex at the free end of beaim.^ 
[When the ordinates are laid off equally on each side af the; 
centre line, the nature of the curve is not changed, but a pair 
of parabolas are obtained, as in fig. 12] 

When the cross sections are similar figures or when 

h^=nd, then knd^'=^x^ . . . . No. ISt 



Here d varies as the cube root of the square of tlie 
length y^ , and the lines of the top, bottom and sides of 



* This may also be obtained by direct integration, ql xdx 



qx^ 



^ 



gxdx being the moment of a very small portion of the length. 
4 



38 

beam are semi-cubical parabolas, with the vertices at the 
free end of beam. {See Fig, 13.) 

Case od, — Beam supported at both ends, and loaded in 
the centre, or at any intermediate point. When the weio-ht 
is on the middle of beam, each support receives one-half of 
it, and the moment of rupture — 

W L JVL 

=—y^—= =Jcbd' No. 21 

2 2 4^^ 
When the weight is at any intermediate point, say at a 

distance x, from one support {see Fig. 14), then the pres- 

(L — X \ X 
I W. and P2 =—W. 

The moment of rupture =Pi x= P<^ {L — x)=kbd'^ ; also 

(L — X \ -Lx — X ^ 
— \w= ^W=kbd\ - No. 22. 

And the strength of the beam under these circumstances 
varies inversely as the product of the two segments in 
which its length is divided, viz., x and L — x. 

By inspecting Figs. 5 and 6, it will be readily seen that 
Case Sd is but a variation of Case 1st, the same shapes be- 
ing necessary for beams similarly circumstanced. For 
instance, when the depth is constant the sides are straight 
lines, the plan of the beam being two triangles with their 
bases abutting each other at the point of application of the 
weight. {See Fig. 15.) 

When the breadth is constant the depth varies as the 
ordinates of the common parabola, {see Fig. 14), and when 
the cross sections are similar figures, the curved sides are 
cubic parabolas. 

Case ith, — Beam supported at both ends, and uniformly 
loaded. 

{See Fig 18 for illustration of this case.) 



39 

The moment of rupture for any section at a distance x 
from a support, may be d'"etermined as follows : — 

Let g= weight on a unit of length, the qL = W, and ths 

W qL 

pressure at each support= — = — . It will be seen by in- 

2 2 

W 

•specting the figure that — (or the resistance of support), 

acting at the end of the arm x, tends to break the beam in 
one direction, while the weight which lies between the sup- 
port and a—:/, and which is equal to qx, acts at its centre 

X ^ ^ ^ 

of ^-ayity on an arm — , and tends to break the beam in the 

A 

opposite direction ; hence the actual moment of rupture 
must be the difference between these two moments ; that is, 

hhd?=:^i~y.x\—iqx X —X^-iLx—oA, - No. 23. 

When the depth is constant, 6= f Lx — .^^ J 

IkdA / 

=^'This is an equation to the common parabola, the origin of 

the co-ordinates being at one of the supports ; and the 

vertex (or vertices of a pair, when the ordinates are laid off 

equally on each side of the centre line of beam) at the 

middle of the length. {See Fig. 17) 

When the breadth is constant, d=^\\ x ,/ L,oc — : 

» 2Jcb V 

t This is an equation to an ellipse ; the origin of the 
co-ordinates being at one of the supports. {See Fig, 16.) 



* Vide Loomis' Analytical Geometry, Art. 128. 
t '' '* " '' Art. 70. 



-x' 



40 

WKen all the sections are similar figures, knd^— i{Lz — x') 

No. 24, 



]'^ — ; 

2kn V 



^rovisio7i to be made at. the Ends of the Beams for the 
direct Shearing Force of the Weight. 

The force tending to shear oif the beams at the ends is 
'flirectlj as the weight or load, and if we neglected the 
^?reight of ihe beam itself, it remains the same for ^ny 
length of beam. 

^Representing by f the shearing force of one square inch 

c*f the material, then for beams fixed at one end and loaded 

a^ the other, the area of cross section at the free end should 

W 
l^e at loast equal to — 

/ 

When the beam is sustained at both ends, the area should 

W 

he — . 
2/ 

It is also generally necessary in practice to modify 

iie ends of the beam, so as to obtain suitable bearing 

isnrface, &c. 



41 



Table of co-efficients of the transverse strength of m^t^^ 
rials having rectangular sections. 



Material. 



Values of K 

in its. 
avoirdupois. 



Values of K 

in tons 
of 2,000 lbs. 



Values of S 
in Tos. or co- 
efficients for 
safe weight 



AutlifiritT.- 



Timber. 

Ash, English.. . 

" American 

" '' Swamp .. 

" Black.... 

Beech, English ...-...- 

" American white. . 

" " red.... 

Birch, common 

'' • American black. . 
" '* yellow. 

Cedar, Bermuda 

*• American white. . 

El-iK, English 

" Canada rock 

Hickory, American 

" " bitter nut 

Oak, English 

" American white . . . 

" « red 

" " live.... 

" Adriatic 

Pine, Virginia 

" Memel. 

" American white. . . 

<' '- red 

" ** yellow.. 

" " pitch . . . 

Spruce 

" American 

Norway spar 

Deal, Christiana ■ - 

Canada, balsam 

Hemlock * 

Larch 

" Amer. or Tamarak 
Lignum Vitae 

Teak 

Mahogany .Span . seasoned 
" Honduras *' 

Chestnut tree 

Walnut, green 

"Willow 

New England pine 

Pitch pine 



168. 
149. 

97. 

71. 
129. 
115. 
144. 
160. 
171. 
111. 
1-20. 

63. 

65. 
164. 
177. 
122. 
141. 
145. 
140. 
155. 
122. 
121. 
112. 
102. 
127. 

98. 
143. 
112, 

86. 
122. 
130. 

93. 

95. 
111. 

75. 
167. 
150. 
141. 
170. 
106. 
159. 
112. 
121. 

91. 

91. 
136. 



.084 
.074 
.048 
.035 
.064 
.057 
.072 
.080 
.085 
.055 
.060 
.031 
.032 
.082 
.088 
.061 
.070 
.072 
.070 
.077 
.C61 
.060 
.056 
.051 
.063 
.049 
.071 
.056 
.043 
.061 
.065 
.046 
.047 
.055 
.037 
.083 
.075 
.070 
.085 
.053 
.079 
.056 
.060 
.045 
.045 
.068 



42. 
37. 
24. 
17. 
32. 
28. 
36. 
40. 
42. 
27. 
30. 
15. 
16. 
41. 
41. 
30. 
35. 
36. 
35. 
38. 
30. 
30. 
28. 
25. 
31. 
24. 
35. 
28. 
21. 
30. 
32. 
23. 
23. 
27. 
18. 
41. 
37. 
35. 
42. 
26. 
39. 
28. 
30. 
22. 
22. 
34. 



Papers of tS«:' 
Royal Engi- 
neers, Vol. '^- 



Barlow. 
\ Tredgold,. 

j Ebbeisi 

Tredgol<^ 
\ BarloWv 



42 



Values 
of Ji 
in lbs. 



Values 
of S£ in 
tons of 
2 0G01bs 



Values 
of ;§? 
in lbs. 



AMERICAN STONE. 

Blue Stone flagging 

Quincy granite 

Little Falls freestone 

Belleville, N.J. freestone. . . . 
Granite bl. (another quarry) 

Connecticut freestone 

Dorchester do. 

Aubigny do. ■ 

Caen do. 



31. 


.015 


7.7 ^ 


26. 


.013 


6.5 


24. 


.012 


6. 


20. 


.010 


5. 


18. 


.009 


4.5 


13. 


.0065 


3.2 


11. 


.0055 


2.5 


9. 


.0045 


2.2 


6. 


.003 


1.5 J 



From experiments 

made by R. G. Hat- 

)^ field, Architect New 

York. Jour. Frank. 

Inst. 1858. 



CAST IRON. 

Ponkey, No. 3, cold blast. . . 

Devon, No. 3, hot blast 

Carron, No. 3, " " 

Butterley, . . 

Buflfery, No. 1, cb 

Devon, No. 3, c. b 

Coedtalon, No. 2, h. b. 

Do. do. c. b 



Common pig iron : . . . 
Good common castings 



664. ^ 


.327 


163. 


604. 


.302 


151. 


592. 


.296 


148. 


564. 


.282 


141. 


520. 


.260 


330. 


504. 


.252 


126. 


468. 


.234 


117. 


462. 


231 


115. 


520. 


.260 


130. 


610. 


.305 


152. 


540. 


.270 


135. 


500. 


.250 


125. 


625. 


.312 


156. 



I E. Hodgkinson, 

y and 

I W. Fairbairn. 



W.H. Barlow, 1858. 
Maj.Wade, U.S. A. 



WROUGHT ISDN. 



Swedish bar 

Hammered iron bar 

Hammered puddled steel bar 



798. 


,399 


199. 


714. 


.357 


178. 


714. 


.357 


178 


577. 


.288 


144. 


750. 


.375 


187. 


1869. 


.934 


467. 


2709. 


1 354 


677. 



Drewry.' 

Lillie. 

Edwin Clark. 

Barlow. 

General Millar. 

i Mersey steel & iron 

i works, Eng. 1857. 



Relative Strength of Wrought and Cast Iron Beams, 

Mr. Hughes, in his paper before referred to, remarks as 
to the transverse strength of wrought iron : — 

*' In comparing this with cast iron, it is important to 
observe the remarkable distinction which characterises the 
two separate forms of iron. Cast iron bends but little be- 
fore it breaks, and its elasticity is injured by a very small 
deflection. "Wrought iron, on the contrary, will bend con* 
siderably without breaking, and will bend much more than 
cast iron before its elasticity is destroyed. 

" Owing to this distinction the breaking weight of cast 
iron is usually considered, and the safe weight assumed as 



43 



a certain fixed proportion of the breaking weight ; but in 
wrought iron the breaking weight is seldom determined. It 
is sufficient to ascertain the weights which produce certain 
deflections, and particularly that which injures elasticity ; 
the safe permanent or occasional load must be considerably 
less than the latter." 

Taking the results of numerous experiments, and the 
practice of the best engineers in Europe and this country, 
as our guide — we may adopt as co-efficients of safe weights 
for beams hayins^ rectanorular sections — 

For cast iron, S= 100 



And for wrought iron, S=150 



in lbs. 



Example. — A weight of 1,029 lbs. is to be supported at 
the end of a beam 7 feet long and 8 inches deep. How 
thick should it be to sustain the above weight safely, if 
made of cast iron ; and how thick if made of wrought 
iron? For this description of beam, WL—SAd=Sb(Pj and 
WL 



Sd'' 

TF= 1,029 lbs 
L= 7 feet. 
d=8 feet. 
For cast iron, 
*' wrought ** 



S=100. 
5=150. 



Thickness of cast iron 
beam — 
1,029X7 



b=- 



=ir 



100x64 
Thickness of wrought iron 
beam — 
1,029x7 



6 = 



150x64 



S" 
4 



44: 



Experimeiits of Mr, Macperson, of Philadelphia, Fenn., on 
the Transverse Strength of certain Alloys, 1855. 

All the bars were cylindrical, and of the following 
dimensions : — Extreme length, 25 inches, and turned to a 
diameter of 2 ^-^ inches. 

The weights were applied in the centre ; the distance 
between the supports was 24 inches. 

The formula for the breaking weight of beams, supported 

4:hAd 

at both ends and loaded in the middle, isT'F'^:- . [See No, 



WL 



Hence, k=^ = 

4Ad 4X3.34X2.0625 
L=2. 
J.= 3.34 square inches. 

^=2.0625'' 



L 

W= 0764W 



No. 



Nature of Mixture, 



Breaking 
weight. 



Ultimate 
deflection 
in inches. 



Values of 
k in lbs. 



Values of 

«-4 



10 lbs. Copper, 10 oz, Tin, 12 oz. Zinc, 8 cz. 
Lead 

6 lbs. Copper, 4 lbs. Zinc . . , . . 

2 " " 1 " " 

9 " " 1 " Tin '. 

10 " " 1 " " 4oz. Zinc........ 

Same as No. 5, but cast some weeks later, 

it endured without breaking , . 

Syz lbs. Copper, 8 oz. Tin, 1 lb. Zinc 

Same as No. 1, with the addition of 1 lb. 
of Tin to every 12 lbs. of mixture .... 
A piece of refined bar Iron of the same 
length and di^.m deflected slowly to 4 
inches and sustained 12375 lbs. and by 
increasing to 17000 it deflected to 5)^. 



6874 


7^4" 


525. 


6489 


2. ■ 


495. 


6158 


7 9-16 


470. 


7802 


414 


596. 


10530 


'ryz 


804. 


11695 


1% 


893. 


6461 


5H 


493. 


5333 


i^i 


407. 


17000 


b% 


1298. 



131. 
123. 
117. 
149. 
201. 

223. 
123. 

101. 



324. 



Experiments on Cast Iron Beams, having Cross Sections 
of Varioiis Forms. • 

In the valuable experiments by Messrs. Hodgkinson and 
Fairbairn, several theories that had been sanctioned by high 
authority were shown to be erroneous. *' It had been held 
by Tredgold and others, that a beam of the T section was 
equally strong whether the rib or the flange was uppermost 



45 

Tbut Messrs. H. and F. found that in a beam of this shape, 
supported at both ends with the rib downwards, the break- 
ing weight was 266 lbs., and when the rib was placed 
uppermost, the breaking weight was 1,050 lbs., or nearly 4 
times as strong as before. It had also been held by earlier 
writers, that the elasticities of cast iron beams would remain 
perfect until loaded with nearly one-third of the breaking 
weight. But Mr. Hodgkinson found, by his experiments, 
that a visible set was produced in some cases with a weight 
not exceeding one-fiftieth to one-eightieth of the breaking 
weight, and concludes that there is no permanent weight, 
however small, that will not injure the elasticity of cast 
iron." He also ascertained from a large number of care- 
fully conducted experiments, that the beam of greatest 
strength, with a given amount of material, was obtained by 
making the flange which is exposed to tensile strain six 
times the area of the flange subject to compression, the two 
flanges being connected with a thin vertical rib. This result 
has since been shown theoretically, reasoning from the now 
well-known fact, that cast iron offers about six times the 
resistance to compression that it does to extension. 

The accompanying table contains some of the more 
important experiments of Messrs. Hodgkinson, Barlow and 
Fairbalrn, and the co-efficients deduced from them exhibit 
the relative advantage of the various sections. 

Prof. Barlow's experiments show that large sections are 
proportionately weaker than small ones. 

It should not be forgotten that, in large beams, one -half of 
the weight of the beam itself should be added to the amount 
which is required to be borne. This becomes a very impor- 
tant item in designing girder bridges. Of course the exact 
weight canrot be known until the section is determined, but 
for all practical purposes it may be generally approximated. 



46 



Experiments on the 
of cast iron beams of various sections, 



Shape and position of 
cross section. 



X 


d 


•^ 


Length 


Depth 


Total area 


or clear 


of beam 


of cros? 


span in 


in 


section in 


ft, & ins. 


inches . 


sq. in's. 



Ratio of 


Ratio of bottom 


ttom flange 


flange to the 


to top 


total area of 


flange. 


cross section. 





^m^ 



Usually called 
the Tredgold 
beam. 



Sometimes called 

the Fairbaira 

beam. 




The 

Hodgkinson 

beam 



5' 0" 


1.01 
2. 


1.032 
4.01 


»» ') 


1.44 


1.04 


» JJ 


2.83 


4.02 


» 1i 


1.12 


.989 


J) >J 


2.5 

2.2 


4.97 
3.78 


4' 6" 


5li 


2.82 


j> >? 


M 


3.2 


» >j 


J) 


2.87 


19' 0" 
22' 0" 
1 27.3' 


» 
?) 

22. 
17. 
18. 


3.02 

3.37 

4.5 

5.0 

6.4 

6.5 

5.41 
39.6. 
24 5* 
29.1 



1 :1 



2 ; 1 



1 ; 4 



1 : 2.73 



1 ; 3 



4 


. J 




4J^ 






4 






b}i 






6 






6.73 






6 72 






2.75' 






4.9 : 






4.6 : 







2.5 

2.15 

2 

1.73 

4.5 

1.51 

1.62 

35 

2 

2 



47 



Transverse strength 
[supported at both ends and loaded at the centre ] 



w 

Breaking 

weight in lbs. 
avoirdupois. 



Values of k in lbs. 
in the formula 



W= 



Values of k in tons 
of 2,000 lbs. in the 
formula 
ikAd 



W^ 



Values of S or co- 
efficients of safe 
weight in lbs. 
k 
4 



S=: 



Authority. 



52' 
347^ 



449. 

2988. 
474. 



4143. 
3132. 



6678. 



6270. 



r368. 



8270. 
10727. 
J4462. 
16730. 
26084. 
23249. 
21009. 
116480. 
50064. 
65632. 



658. 
542. 



374. 

330. 
538. 



417 
471 



520. 



568. 



564. 



602. 
700. 
707. 
736. 
896. 
786. 
854. 
635. 
670. 
855. 



.329 
.271 



.187 

.165 
.269 



.208 
.235 



.260 



.284 



.282 



.301 

.35 

.353 

.368 

.448 

.393 

.427 

.317 

.335 

.427 



164. 
135. 



93, 

62. 
134. 



104, 
117. 



130. 



142. 



141. 



150. 
175. 
176. 
184. 
224. 
196. 
213. 
1^8. 
167. 
213. 



W. H. Barlow. 

See Civil 
Engineers an(J 

Architects' 
Journal, 1858, 



E.Hodgkinson 

See Artizan 

Journal, 

1857-8. 

Hughes' 

Papers. 



"W. Fairbairn. 
See Artizan Jour. '57-8 



48 

Example — A '* Hodgkinson" beam, supported at both 
ends, has an area of cross section at the centre of 20 square 
inches ; the bottom flange having 12 square inches, and the 
top flange 2 square inches, or the ratio to each other as 6 
to 1. 

The depth of the beam is 15 inches, and the distance 
between the supports 18 feet. What is the breaking weight 
at the centre in tons of 2,000 lbs '? 

For beams supported at both ends, and loaded at the 

AkAd 

centre, we have, W— — 

L 

J.-20, i=18, and ^-15. 

Eeferring to the table, we find for a beam proportioned as 

above, k= .448. 

4X.448x20xl5 

Hence, TF-- :=.29.8 tons of 2,000 lbs'. 

18 



Experiments Mpported at both ends and 



Shape and position 3 




Rolled Beam 



•3)$"M 



of 
, in 
Ad 



k 

values of 
k in tons 
of 2000 lbs 



in lbs co- 
efl&cient of 
safe w'gts. 



Authority. 



.337 



.337 
.404 



168. 



168. 
202. 



,301 150. 



W. Fairbairn, 



F( 



Experiments on the Transverse Strength o/Weought Iron beams of various sections, supported at both ends and 

loaded dt tJie centre. 




Tliomas Davies, 

S. See 

I Civil Engineer 

anil AicliitecU' 

\ Journal, 1857— 

\ page 20. (As all 

,tlie beams which 

/llr. Davies ei- 

/ per!in»nted upon 

were intended for 

actual uae.theul- 

timalt strength* 

were not ascer 

tained.) 



51 



)-efficie7its for the Transverse Strength of Iron Beams 
from Edivin Clark's icork on the Conicay and Britannia 
Bridges, 



CAST IRON. 






1 


Values of A; 


in Ibs^ 


liform rectangular tubes, 
j*' elliptical " - 


- 


511 


- 


664 


1 " circular ** - 


- 


53a 


\ " square " 


- 


637 


iiall rectangular bars, 


- 


634 


rge- 


- 


420; 


lall round bars, - 


- 


sm 



WU OUGHT IRON. 

ctangular tubes, with thick top and bottom plates^^, 
and thin sides, as in the Britannia and Conwaj 

bridges, • - - - - 1,246^ 

ctangular welded tubes, uniform thickness, - 1,096. 

liptical '' '' '' " - 1,040 

rcular " " " " - 975 

und rivetted tubes, . . . . 608 

al '* .... 714 

octangular '' .... 84a 

tw rectangular bars (solid), - - - 714 

Warren or Half Lattice Girders. 

This variety of lattice girder seems justly to obtain at 

esent the preference of engineers. Theoretically, it has 

een declared to be the most perfect beam, giving the 

laximum of strength with a minimum of material. Owing 

practical considerations, however, the Fairbairn or 

wrought iron plate beam still continues a formidable rival. 

For a full investigation of the subject, see Prof. Eankine's 



52 

''Applied Mechanics," London, 1858, and the Civil Engineer 

and Architect's Journal for 1853 and '55. 

We propose here to furnish only some of the more 

prominent facts. 

There are four cases which present themselves — 

1st, When a beam is fixed at one end, and loaded at the 

other. 

2d. The same beam uniformly loaded. 

3d. When a beam is supported at both ends, and loaded 

in the centre. 

4th. Beam supported at both ends, and uniformly loajded 

LetjL=The total length of unsupported part of beam, or 

the clear span. 

Let c^== Depth of beam. 

Let ^= Number of diagonals, as 1, 2, 3, 4, &c., (see Fig.) 

Let ^— Angle contained between the diagonal and a vertical 

line. 

Let !¥= Total weight. 

Let /S= Horizontal strain (on either the upper or lower 

strips). 

Let 2— Strain on a diagonal. 

Case 1st. Beam fixed at one end, and loaded at the other. 

(See Fig. 20.) 

The horizontal strains are greatest at the point of support, 

and decrease directly as the distance from that point, or 

increase directly as the distance from the weight. 

At any point distant n diagonals from the weight, S= 

L 

W, 71, tang. <I>, but as n, tang. ^xl — L, and n tang. <l>— — . 

d. 

We may substitute this latter value in the equation first 

given, and we obtain — 

WL 
5- ^ No. 25 

d 



53 

It will tlius be noted that the horizontal strains do not 
depend on any values of $ ; they are the same for any 

angle. 

The strain on each diagonal remains constant throughout 
thebeam, and S=TT^. secants. - - " No. 26 

Case 2d. Beam fixed at one end, and wiiformlij loaded. 

Kepresenting by q, the weight per unit of length, and by 
X any given distance from the free end of beam. Then the 

q.x" ^ _ 

horizontal strain at a distance x, will be 8=-—-, that is, the 

2. a 

horizontal strain increases towards the support as the square 
of the leno-th ; the other elements being constant. When x 
becomes e°qual to L, then qx^^=W.L. Hence, at the point 

„ , c. , . . - - - No. 27 

of support, 0= , - 

■ ■ 1.d 

The strain on any diagonal increases directly as its 
distance from the free end of beam. We have, 
S=?.a;. secant <!>, - - - " " - No. 28 

Case 3d. Beam supported at both ends, and loaded at the 

centre. • . 

This is but a variation of case 1st, as was noticed in 

regard to solid beams. 

The horizontal strain is greatest at the centre, and 
decreases toward the supports directly as the length. 

S=. =strain at the centre, - - - - INO, ^J 

Ld 
The strain on the diagonals is constant, and 

S,=— . secant $, No. 30 

2 
Case 4th. Beam supported at both ends, and uniformly 
loaded. 



54 

q being the weight per unit of length; the horizontal 

strain at any point at a distance x from one of the supports 

q.{Ij X — x^) 
is 5'= , - - - - - - No. 81 

When X becomes equal to one-half the length of beam, 

L q ,V V\ q V- 

or=— , then S^ — I — i = — x — , but qL^ W, hence 

2 2.d\2 4/ 2.d 4: 

WL 

^= —horizontal strain at the centre of beam. 

The strain on the diagonals is least at the centre, and in- 
*0reases toward the supports directly as the distance from 
the centre. 

2-=l ::^; I ^. secant ^, - - - - No. 32 



=(__.),. 



In illustration of this fact, see Fig. 21, where the strains 
are represented by arrows in magnitude and direction, and 
ihe weight by balls suspended from each joint. 

The facts embodied in the foregoing formulas for Warren 
^girders, were also ascertained experimentall}^ by W. T. 
Doyne and W. B. Blood, civil engineers, who constructed 
models for the express purpose of discovering the agree- 
ment between theory and fact. (See Civil Engineer and 
Architect's Journal, 1853, p. 20 ) 



Trussed Cast Iron Girder. 

Mr. Fairbairn and others have investigated and experi- 
mented upon this form of girder. The results obtained 
show that there is little, if any, advantage over the simple 
beam, while the combination deceives by leading persons to 
make false estimates of its strength. 



55 

Owing to the different rates of extension of wrought and • 
cast iron, it commonly happens that, in trussed girders, the 
casting breaks before the tension rods are fully strained. 
Again, if the tension rods are highly strained at first, they 
generally yield with weights which might be borne by the 
beam itself. Mr. Fairbairn found the most favorable condi- 
tions of strength to be when the tension rods have a mode- 
rate initial strain upon them ; but the gain, even then, is too 
small to warrant the use of the tension rods at all. He 
found, also, that the cost of the trussed, compared with the 
simple girder of equal strength, was as one to two thirds ;- 
that is, the trussed girder was about 50 per cent, more- 
costly. [See Fairbairn's Treatise on wrought and cast iron? 
1854.] 

About ten or twelve years ago, in England, there were- 
numerous failures of trussed cast iron girders ; one of the - 
most remarkable was that of the bridge across the river 
Dee, on the line of the Chester and Holyhead railway. It 
consisted of three spans, each 98 feet in the clear ; it fell 
under a passing train in 1847. This and other accidents of 
a similar nature diminished the confidence of engineers in 
such combinations, and we are informed by English writers 
that there have been scarcely any bridges of this descrip- 
tion built within the last ten years ; the wrought iron plate 
and lattice girders are now rapidly sjiperseding other 
varieties. 

The account of the following interesting experiment, 
which was made at Mr. Cubitt's works on the Thames, we 
extract from the London Builder of 1848 : 

" The girder was of cast iron, 28 feet long, 10 inches 
deep, parallel, and of the same section throughout. A 
strong frame was cast upon the ends of it, in order that the 
inclination of the tension rods mio'ht be varied, or, rather,, 



56 



that the ends of them might be raised to different heights 
above the bottom fiange. [See Fig. 19.] 

" The tension rods were of wrought iron, one inch in 
diameter — one on each side of the girder attached at the 
points BB to iron pins two inches in diameter, which 
passed under the bottom flange ; at the points AA they 
were connected by means of plates clipping the ends of the 
girder, and tightened up by nuts and screws. 

*' A weight of 8,960 lbs, was placed upon the centre of 
the girder, and the deflection (2|") carefully taken. 

'* The tension rods were then put on and applied in 
various places, and the weight required to produce the 
same deflection is given in the following table ; the distance 
of the bearings was in all cases 27 feet : — 



No. of 
experi- 
ment. 



Distance of points 
A A above the ■bot- 
tom of girder, in inch's 



Distance of points B B 
on centreof pins. JeZow; the 
bottom of girders, in inc's 



Weight required to pro- 
duce the same deflection 
of 2^ inches, in lbs. 



30 

24 
18 
10 
24 
10 
5 



1 
1 
1 
1 

6H 



8848. 

8960. 

9072. 
10192. 

9856. 
13552. 
12544. 



*' After the above experiment had been made, the ten- 
sion rods were removed, and the deflection with 8,960 lbs. 
was found to be the same as at the commencement. 

" Our readers will observe that, strange as the fact may 
appear, when one 'end of each tension rod was 30 inches 
above the bottom of the girder, and the other end one inch 
below the bottom, the same deflection was produced by 112 
lbs, less than when there were no rods. 

When the distance of the points AA above the bottom of 
the girder was lessened to 10 inches, and the distance of 
the points of suspension {BB) beloiv the bottom of the gir- 
der was increased to 6^ inches, the girder was strengthened 
50 per cent." 



DEFLECTION OF BEAMS. 

We give below the general formulas for deflections, from 
Prof. Moseley's *' Mechanics of Engineering," edited by 
Prof. Mahan, 1858, merely changing the letters which he 
has employed, to agree ivith those heretofore used. 
Let i=The total length of unsupported part of beam, or the 

clear span in inches. 
Let J=The moment of inertia of the cross section, in respect 

to an axis passing through its centre of gravity 

(dimensions in inches). 
Let £=The modulus of elasticity of the given material 

in lbs. 
Let e=Deflection in inches. 
Let Vr=The weight which produces the deflection (in lbs.)- 

Then, for a beam fixed at one end and loaded at the other, 
g_ ^ . . . . . . . - No. 33 

For a beam fixed at one end and uniformly loaded, 

g_ ^ No. 34 

%EI 

For a beam supported at both ends and loaded at the 

wn 

centre, e= , No. 35 

48£/ 
For a beam supported at both ends and uniformly loaded, 

5 WV 

e=-.>< , . - No. 36 

8 48^1 

For rectangular sections, I=\-^ hd\ when i=breadth, and 
c^= depth of beam. For I shaped and hollow rectangular 
sections, let b and d remain as before, and let h^ and (^i repre- 
sent the breadth and the depth of hollow part; then 



58 

I=}.{bd^ — bidi). For circular sections, let R be the radius ; 
then I=i—R'=7854:R' {R'—r% 

For annular sections or hollow cylinders, let i^= outer 
radius, and r= inner radius ; then J=.7854 R^, 

The general law of deflections, as embodied in the fore- 
going formulas, and verified by the experiments of Prof. 
Barlow and others, is that they vary directly as the weight 
and cube of the length, and inversely as the breadth and 
cube of the depth, or as the moment of inertia of the cross 
section. 

The deflections of a beam are always proportional to the 
load within the limit of elasticity; beyond that limit they 
increase rapidly, but do not seem to follow any general 
law. 



Deflections of Rectangular Beams. 
Let <^=The breadth in inchest 

Let d^ "• depth *' ( . . . 

, ^ . />As before. 

Let e= " deflection *' C 

Let H^=The weight in lbs. } 

Let i=The length of span mfeet. 

Let C=K co-efficient, according to the i^ture of the material. 

Then, for a beam fixed at one end and loaded at the 'other, 

WU 
e=C , No. 87 

hd^ 
Beam fixed at one end and uniformly loaded, 

6=1 C , No. 38 

bd' 
Beam supported at both ends and loaded at the centre, 

e = l-C , - - No. 89 

"' bd' 



59 
Beam supported at botli ends and uniformly loaded, 

e=^xLC , No. 40 

As the majority of experiments which have been pub- 
lished were made on beams supported at both ends and 
loaded at the centre, we have deduced the values of C, as 

IQebd' 
given below, from the formula C= . 

The deflection of a round beam, whose diameter is equal 
to the side of the square beam, is 1.7 times that of the square 
beam. 

Values of C. 

■V V,- n S^^^o^ -0002 
lor wrought iron, - - - -0=< ^ 

\ to .0003 

^ . ,, { From .00037 

For cast iron, . - - - - C = j ^^ ^^^^ 

TIMBER. C 

Ash, .004 

Beech, .0048 

^^'^' |.0084 

Maple, common, --«---- .0076 

Willow, ..--.... .0124 

Horse chestnut, .0064 

White pine, American, ----- ,0048 

Pitch pine, " --....- .0064 

English oak, .0044 

Mahogany, Honduras, .004 

Spanish, .0052 

Walnut, green, .008 

Cherry, '^ .0052 

Pear,*^ " .0084 



60 

TOSSION. 

The resistance of solid cylindrical shafts to a twisting 
strain increases directly as the cube of the diameter, while 
the strain is increased directly as the weight m the distance 
from the axis of shaft at which it acts. 

Hence, representing by R the distance of the weight or 

pressure from the axis of shaft in feet, by D the diameter of 

the shaft in inches (see Fig. 22), by /S the co-efficient of the 

given material, and by V\^ the breaking pressure or weight 

in lbs., we have — 

D' 
W=S—, No. 41 

R 

' IWR 
and -D= V No. 42. 

D' 
When D=l and R=l, then — =1, S then becomes equal 

R 
to W— that is to say, S represents the breaking weight of a 
shaft one inch in diameter, applied at the end of a lever one 
foot long. 

Torsional Strength of Hollow Shafts. 

Let IF, R^ D, and S, be as above, and let d represent the 

interior diameter of the cylinder in inches, then — 

D'—d' 
W=S , No. 43 

Dr 

Which is a practical formula deduced from Major Wade s 

experiments, by Lieut. Rodman, of the U. S. Ordnance 
department. 

Values of S, 

For wrought iron, S=MO ) re . ^ . ^i j 7 . 

^ r-r^r. I Co-emcients tor the breaJ^inc^ 
'' cast '' 5=560 V . ^ 

^ Ar.r.\ weiorhts. 

'' bronze '' S=460 ) 



61 



Relative torsional strength of cast iron shafts of different 
forms ^ havidg equal areas of cross sections. 

From Maj. V/ade's experiments on shafts whose cross sec- 
tions were 1, 2 and 3 square inches. 



Solid 
Cylinder 


Solid 
Square. 


Hollow cylinders whose interior and exterior diameters are in the 
proportion of 


4 to 10 


5 to 10 


6 to 10 


7 to 10 


8 to 10 


1.0000 


0.8750 


1.2656 


1.4433 


1.7000 


2.0864 


2.7377 



Thefollowi7ig Table exhibits the results ef experiments made 
by Mr, Dunlop at Glasgoiv, on round bars of wrought iron. 



Length of bars 


Diameter of bars 


Breaking weight 


Length of 


Values of 


in inches. 


in inches. 


in lbs. 


lever.' 


^■5 = D3 


2% 


2 


250 


d 


442 


234 


2^ 


384 




476 


3 


2^ 


408 


•+3 


370 


3 


25^ 


700 


O £JD 


476 


4 


3^ 


1170 




483 


5 


3J^ 


1240 




410 


5 


'6% 


1662 


S3 


446 


5 


4 


1938 


!> 


428 


6 


4^4 


2158 


t-H 


398 



Table shoiving the Working Strength of Wrought Iron 
Shafts ; the 2dAth. Qth. andWi, columns containing the 
products of the cube of the diam. in inches^ by the co- 
efficient of SAFE WEIGHT. 



Diam. of 
shaft in 
iaclies. 



D3X125 



Diam. of 
shaft in 
inches. 



D3 X 125 



Diam. of 
shaft in 
inches. 



D3xl25 



Diam. of 
shaft in 
inches. 



D3^125 



% 
H 
% 
% 
1 

1^ 

1^ 

1^4 

2 

2J^ 

1% 

3 

%\ 

4 



1.95 

6.58 
15.6 
30 5 
52.5 
125. 
243. 
421. 
668. 

1000. 

1423. 

1950. 

2598. 

3375. 

4290. 

5350. 

6587. 

8000. 



4^ 


9587. 


8% 


83625. 


nii 


4^ 


11387. 


9 


91125. 


133^ 


m 


13375. 


m 


98875. 


13^ 


5 


15625. 


93^ 


107125. 


14 


5U 


18000. 


9% 


115750. 


UH 


5H 


20750. 


10 


125000. 


14^ 


5^i 


23750. 


WH 


134500. 


14^ 


6 


27000. 


.1^% 


144625. 


15 


6U 


30500. 


\0\ 


155250. 


153^ 


63^ 


34250. 


11 


166375 


15}^ 


&% 


38375. 


UH 


177875. 


16 


7 


42675. 


n% 


190000. 
202750. 


163iS 


7U 


47625. 


\\% 


17 


7H 


52625. 


12 


216000. 


173^ 


7-^ 


58125. 


nM 


229750. 


18 


8 


64000. 


nvz 


244125. 


18^ 


8U 


70125. 


1234 


259000. 


19 


. 8^ 


76750. 


13 


274625. 


20 ' 



290750. 
307500. 
325875. 
343000. 
361625. 
381000- 
401125. 
420875. 
443250. 
465375. 
512000. 
551500- 
614125. 
669875. 
729000. 
791375. 
857375 
1000000. 



Example illustrating tlie use of the above table. 



The pressure on the piston of a steam engine is 23805 lbs. 

and the length of crank 4 feet 6 inches. What diameter 

of shaft is necessary, if of wrought iron 1 Multiplying 

23805 1^ 4 5 we obtain 107122, and looking in the table we 

find the diam. of shaft corresponding to it to be 9 J inches. 

For from formula No. 41, we have 

co-efficient m D^ 

\V= : —, whence T^Ti^^ co-efficient n D^ 

R 
That is, the product of the weight or pressure by the 

length of lever must be equal to the product of the cube of 

the diameter of shaft by the co-efficient. 

Cast Iron Shafts. 

It is customary, owing to the defective elasticity of cast 

iron, to make the di'ameter of cast iron shafts li times that 

' 10 

diameter which would be proper if wrought iron were used. 

Elasticity of Torsion. 
Let D=The diameter of shaft in inches. 
Let i^=The length of crank or lever in inches. 
Let Zr=The length of shaft in inches. 

Let W=ThQ weight at the end of the lever or crank in lbs. 

= The angle in degrees through which the shaft is 

twisted ; that is, supposing one end of the shaft 

fixed, G represents the number of degrees through 

which the opposite or free end is moved. If the 

shaft were twisted a complete revolution to the place 

whence it started, 9 would then be 360 degrees. 

WRL 

For wrought iron, 0= , - • - No. 44 



17500 D' 
WRL 



For cast iron, G= 



10000 D' 
WRL 



And for wood, G= 



218 D* 



No. 45 



No. 46 




f 



litlL of J.Bien 60 Fulton Sti\ Tf.X. 



ILLUSTRATIONS OF THE STRENGTH OF MATERIALS. 




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B,-<nns fi,r,',l al onr ,;,d 






Fig. S. Fig\ 2. Fig 



3 



Figl. y^^^ 




Depth 



u 



Fic,.l7. \ ^ 





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